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Computation

By the finite determinacy theorem, we may assume that , , is a polynomial. Since is faithfully flat and all data will be defined over , we may replace by and, similarly, by and by for the computation. With the additional assumption , all data will be defined over , and we can apply methods of computer algebra. Using standard basis methods for local rings, one can compute a monomial -basis of

Since , represents a -basis of and a -basis of by Nakayama's lemma.

The matrix of with respect to is defined by . Since , we obtain for

So the action of in terms of the -basis is determined by the matrix by the above formula.

A reduced normalform with respect to a local monomial ordering allows to compute the projection to the first summand in

Since and , the matrix of with respect to can be computed up to arbitrarily high order.

The basis representation of with respect to defined by can be computed inductively by

Using standard basis methods, one can check if and compute a -basis of with

Then the matrix of with respect to the -basis of is defined by the formula , and for defined by

Hence, the basis representation of

with respect to is

The basis representation of with respect to is defined by , and is the basis representation of with respect to . The matrix of with respect to the canonical -basis

of is given by the block matrix

where . Since the eigenvalues of are rational, they can be computed using univariate factorization over the rational numbers. Then the V-filtration can be computed using methods of linear algebra.

Next: Summary Up: Algorithm Previous: Idea
Christoph Lossen
2001-03-21